Note: Please solve this problem with MATLAB

We want to investigate transient temperature profile in a rod fin. Initially fin temperature is uniform at surrounding air temperature, Ta$.$ Suddenly its base temperature is raised to T$0$ and is kept constant at this value. Heat is then conducted in the axial direction and dissipated into air via free convection. Suppose temperature is function of time and axial distance, z$,$ only during this transient period. This assumption is valid provided that Bi number is sufficiently small. The followings are also known. Fin material is aluminum. Its length and diameter are $20$ and $1$ cm$,$ respectively. Ta is $20$C and T$0$ is $150$C$.$

Note that local heat transfer coefficient should be employed to take convective heat transfer into account.

$1)$ Obtain the governing differential equation and initial and boundary conditions for the described process.

$2)$ Choose a finite difference numerical scheme to solve this problem. Describe it briefly.

$3)$ What is your time step? Show that after a certain value of time step, unstability occurs in the case of an explicit scheme. $4)$ Present numerical solutions at t $=0\mathrm{.}5$ s$,$ t $=5$ s$,$ t $=60$ s$,$ and t $=5$ min.

$5)$ What are the advantages and disadvantages of the numerical scheme you adopted?

$6)$ Verify that low Bi number assumption is valid.

$7)$ Compare your temperature profile at t$=5$ min with that of analytical result for the steady operation.